Sheaf Topos Theory: A powerful setting for Lagrangian Field Theory

We provide an introductory exposition to the sheaf topos theoretic description of classical field theory motivated by the rigorous description of both $(\mathbf{i})$ the variational calculus of (infinite dimensional) field-theoretic spaces, and $(\mathbf{i}\mathbf{i})$ the non-triviality of classical fermionic field spaces. These considerations naturally lead to the definition of the sheaf topos of super smooth sets. We close by indicating natural generalizations necessary to include to the description of infinitesimal structure of field spaces and further the non-perturbative description of (higher) gauge fields.